Ray Tracing: The Rest of Your Life Peter Shirley Version 1.123

Copyright 2018. Peter Shirley. All rights reserved.

Chapter 0: Overview In Ray Tracing In One Weekend and Ray Tracing: the Next Week, you built a “real” ray tracer. In this volume, I assume you will be pursuing a career related to ray tracing and we will dive into the math of creating a very serious ray tracer. When you are done you should be ready to start messing with the many serious commercial ray tracers underlying the movie and product design industries. There are many many things I do not cover in this short volume; I dive into only one of many ways to write a Monte Carlo rendering program. I don’t do shadow rays (instead I make rays more likely to go toward lights), bidirectional methods, Metropolis methods, or photon mapping. What I do is speak in the language of the field that studies those methods. I think of this book as a deep exposure that can be your first of many, and it will equip you with some of the concepts, math, and terms you will need to study the others. As before, www.in1weekend.com will have further readings and references. Acknowledgements: thanks to Dave Hart and Jean Buckley for help on the original manuscript. Thanks to Paul Melis, Nakata Daisuke, Filipe Scur, Vahan Sosoyan, and Matthew Heimlich for finding errors.

Chapter 1: A Simple Monte Carlo Program

Let’s start with one of the simplest Monte Carlo (MC) programs. MC programs give a statistical estimate of an answer, and this estimate gets more and more accurate the longer you run it. This basic characteristic of simple programs producing noisy but ever-better answers is what

MC is all about, and it is especially good for applications like graphics where great accuracy is not needed. As an example, let’s estimate Pi. There are many ways to do this, with the Buffon Needle problem being a classic case study. We’ll do a variation inspired by that. Suppose you have a circle inscribed inside a square:

Now, suppose you pick random points inside the square. The fraction of those random points that end up inside the circle should be proportional to the area of the circle. The exact fraction should in fact be the ratio of the circle area to the square area. Fraction = (Pi*R*R) / ((2R)*(2R)) = Pi/4 Since the R cancels out, we can pick whatever is computationally convenient. Let’s go with R=1 centered at the origin:

This gives me the answer Estimate of Pi = 3.196

If we change the program to run forever and just print out a running estimate:

We get very quickly near Pi, and then more slowly zero in on it. This is an example of the Law of Diminishing Returns, where each sample helps less than the last. This is the worst part of MC. We can mitigate this diminishing return by stratifying the samples (often called jittering), where instead of taking random samples, we take a grid and take one sample within each:

This changes the sample generation, but we need to know how many samples we are taking in advance because we need to know the grid. Let’s take a hundred million and try it both ways:

I get: Regular Estimate of Pi = 3.1415 Stratified Estimate of Pi = 3.14159 Interestingly, the stratified method is not only better, it converges with a better asymptotic rate! Unfortunately, this advantage decreases with the dimension of the problem (so for example, with the 3D sphere volume version the gap would be less). This is called the Curse of Dimensionality. We are going to be very high dimensional (each reflection adds two dimensions), so I won't stratify in this book. But if you are ever doing single-reflection or shadowing or some strictly 2D problem, you definitely want to stratify.

Chapter 2: One Dimensional MC Integration Integration is all about computing areas and volumes, so we could have framed Chapter 1 in an integral form if we wanted to make it maximally confusing. But sometimes integration is the

most natural and clean way to formulate things. Rendering is often such a problem. Let’s look at a classic integral:

In computer sciency notation, we might write this as: I = area( x^2, 0, 2 ) We could also write it as: I = 2*average(x^2, 0, 2) This suggests a MC approach:

This, as expected, produces approximately the exact answer we get with algebra, I = 8/3. But we could also do it for functions that we can’t analytically integrate like log(sin(x)). In graphics, we often have functions we can evaluate but can’t write down explicitly, or functions we can only probabilistically evaluate. That is in fact what the ray tracing color() function of the last two books is-- we don’t know what color is seen in every direction, but we can statistically estimate it in any given dimension.

One problem with the random program we wrote in the first two books is small light sources create too much noise because our uniform sampling doesn’t sample the light often enough. We could lessen that problem if we sent more random samples toward the light, but then we would need to downweight those samples to adjust for the over-sampling. How we do that adjustment? To do that we will need the concept of a probability density function. First, what is a density function? It’s just a continuous form of a histogram. Here’s an example from the histogram Wikipedia page:

If we added data for more trees, the histogram would get taller. If we divided the data into more bins, it would get shorter. A discrete density function differs from a histogram in that it normalizes the frequency y-axis to a fraction or percentage (just a fraction times 100). A continuous histogram, where we take the number of bins to infinity, can’t be a fraction because the height of all the bins would drop to zero. A density function is one where we take the bins and adjust them so they don’t get shorter as we add more bins. For the case of the tree histogram above we might try: Bin-height = (Fraction of trees between height H and H’) / (H-H’) That would work! We could interpret that as a statistical predictor of a tree’s height: Probability a random tree is between H and H’ = Bin-height*(H-H’) If we wanted to know about the chances of being in a span of multiple bins, we would sum.

A probability density function, henceforth pdf, is that fractional histogram made continuous. Let’s make a pdf and use it a bit to understand it more. Suppose I want a random number r between 0 and 2 whose probability is proportional to itself: r. We would expect the pdf p(r) to look something like the figure below. But how high should it be?

The height is just p(2). What should that be? We could reasonably make it anything by convention, and we should pick something that is convenient. Just as with histograms we can sum up (integrate) the region to figure out the probability that r is in some interval (x0,x1): Probability x0 < r < x1 = C * area(p(r), x0, x1) where C is a scaling constant. We may as well make C = 1 for cleanliness, and that is exactly what is done in probability. And we know the probability r has the value 1 somewhere, so for this case area(p(r), 0, 2) = 1. Since p(r) is proportional to r, i.e., p = C’ r for some other constant C’ area(C’r, 0, 2) = INTEGRAL_0^2 C’r dr = (½)C’r^2_0^2 C’r = 2C’ So p(r) = r/2. How do we generate a random number with that pdf p(r)? For that we will need some more machinery. Don’t worry this doesn’t go on forever!

Given a random number from d = drand48() that is uniform and between 0 and 1, we should be able to find some function f(d) that gives us what we want. Suppose e = f(d) = d^2. That is no longer a uniform pdf. The pdf of e will be bigger near 0 than it is near 1 (squaring a number between 0 and 1 makes it smaller). To take this general observation to a function, we need the cumulative probability distribution function P(x): P(x) = area(p, -infinity, x) Note that for x where we didn’t define p(x), p(x) = 0, i.e., the probability of an x there is zero. For our example pdf p(r) = r/2, the P(x) is: P(x) = 0 : x < 0 P(x) = ¼ x^2 : 0 < x < 2 P(x) = 1 : x > 2 One question is, what’s up with x versus r? They are dummy variables-- analogous to the function arguments in a program. If we evaluate P at x = 0.5, we get: P(1.0) = ¼ This says the probability that a random variable with our pdf is less than one is 25%. This gives rise to a clever observation that underlies many methods to generate non-uniform random numbers. We want a function f() that when we call it as f(drand48()) we get a return value with a pdf ¼ x^2. We don’t know what that is, but we do know that 25% of what it returns should be less than 1, and 75% should be above one. If f() is increasing, then we would expect f(0.25) = 1.0. This can be generalized to figure out f() for every possible input: f(P(x)) = x That means f just undoes whatever P does. So, f(x) = P-1(x) The -1 means “inverse function”. Ugly notation, but standard. For our purposes what this means is, if we have pdf p() and its cumulative distribution function P(), then if we do this to a random number we’ll get what we want:

e = P-1(drand48()) For our pdf p(x) = x/2, and corresponding P(x), we need to compute the inverse of P. If we have y = ¼ x^2 we get the inverse by solving for x in terms of y: x = sqrt(4y) Thus our random number with density p we get by: e = sqrt(4*drand48()) Note that does range 0 to 2 as hoped, and if we send in ¼ for drand48() we get 1 as desired. We can now sample our old integral I = integral(x^2, 0, 2) We need to account for the non-uniformity of the pdf of x. Where we sample too much we should down-weight. The pdf is a perfect measure of how much or little sampling is being done. So the weighting function should be proportional to 1/pdf. In fact it is exactly 1/pdf:

SInce we are sampling more where the integrand is big, we might expect less noise and thus faster convergence. This is why using a carefully chosen non-uniform pdf is usually called importance sampling. If we take that same code with uniform samples so the pdf = ½ over the range [0,2] we can use the machinery to get x = 2*drand48() and the code is:

Note that we don’t need that 2 in the 2*sum/N anymore-- that is handled by the pdf, which is 2 when you divide by it. You’ll note that importance sampling helps a little, but not a ton. We could make the pdf follow the integrand exactly: p(x) = (3/8)x^2 And we get the corresponding P(x) = (⅛)x^3 and P-1(x) = pow(8x,⅓) This perfect importance sampling is only possible when we already know the answer (we got P by integrating p analytically), but it’s a good exercise to make sure our code works. For just 1 sample we get:

Which always returns the exact answer. Let’s review now because that was most of the concepts that underlie MC ray tracers.

1. You have an integral of f(x) over some domain [a,b] 2. You pick a pdf p that is non-zero over [a,b] 3. You average a whole ton of f(r)/p(r) where r is a random number r with pdf p. This will always converge to the right answer. The more p follows f, the faster it converges.

Chapter 2: MC Integration on the Sphere of Directions In our ray tracer we pick random directions, and directions can be represented as points on the unit-sphere. The same methodology as before applies. But now we need to have a pdf defined over 2D. Suppose we have this integral over all directions: INTEGRAL( cos^2(theta) ) By MC integration, we should just be able to sample cos^2(theta) / p(direction). But what is direction in that context? We could make it based on polar coordinates, so p would be in terms of (theta, phi). However you do it, remember that a pdf has to integrate to 1 and represent the relative probability of that direction being sampled. We have a method from the last books to take uniform random samples in a unit sphere:

To get points on a unit sphere we can just normalize the vector to those points:

Now what is the pdf of these uniform points? As a density on the unit sphere, it is 1/area of the sphere or 1/(4*Pi). If the integrand is cos^2(theta) and theta is the angle with the z axis:

The analytic answer (if you remember enough advanced calc, check me!) is (4/3)*Pi, and the code above produces that. Next, we are ready to apply that in ray tracing! The key point here is that all the integrals and probability and all that are over the unit sphere. The area on the unit sphere is how you measure the directions. Call it direction, solid angle, or area-- it’s all the same thing. Solid angle is the term usually used. If you are comfortable with that, great! If not, do what I do and imagine the area on the unit sphere that a set of directions goes through. The solid angle omega and the projected area A on the unit sphere are the same thing.

Now let’s go on to the light transport equation we are solving.

Chapter 3: Light Scattering In this chapter we won't actually implement anything. We will set up for a big lighting change in our program in Chapter 4. Our program from the last books already scatters rays from a surface or volume. This is the commonly used model for light interacting with a surface. One natural way to model this is with probability. First, is the light absorbed? Probability of light scattering: A Probability of light being absorbed: 1-A Here A stands for albedo (latin for whiteness). Albedo is a precise technical term in some disciplines, but in all uses it means some form of fractional reflectance. As we implemented for glass, the albedo may vary with incident direction, and it varies with color. In the most physically based renderers, we would use a set of wavelengths for the light color rather than RGB. We can almost always harness our intuition by thinking of R, G, and B as specific long, medium, and short wavelengths. If the light does scatter, it will have a directional distribution that we can describe as a pdf over solid angle. I will refer to this as its scattering pdf: s(direction). The scattering pdf can also vary with incident direction, as you will notice when you look at reflections off a road-- they become mirror-like as your viewing angle approaches grazing. The color of a surface in terms of these quantities is: color = INTEGRAL A * s(direction) * color(direction) Note that A and s() might depend on the view direction, so of course color can vary with view direction. Also A and s() may vary with position on the surface or within the volume. If we apply the MC basic formula we get the following statistical estimate:

Color = (A * s(direction) * color(direction)) / p(direction) Where p(direction) is the pdf of whatever direction we randomly generate. For a Lambertian surface we already implicitly implemented this formula for the special case where p() is a cosine density. The s() of a Lambertian surface is proportional to cos(theta), where theta is the angle relative to the surface normal. Remember that all pdf need to integrate to one. For cos(theta) < 0 we have s(direction) = 0, and the integral of cos over the hemisphere is Pi. To see that remember that in spherical coordinates remember that dA = sin(theta) dtheta dphi, so area = INT(phi from 0 to 2PI) INT(theta from 0 to PI/2) cos(theta)sin(theta) = 2PI * (½) = PI. So for a Lambertian surface the scattering pdf is: s(direction) = cos(theta) / Pi If we sample using the same pdf, so p(direction) = cos(theta) / Pi, the numerator and denominator cancel out and we get: Color = A * s(direction) This is exactly what we had in our original color() function! But we need to generalize now so we can send extra rays in important directions such as toward the lights. The treatment above is slightly non-standard because I want the same math to work for surfaces and volumes. To do otherwise will make some ugly code. If you read the literature, you’ll see reflection described by the bidirectional reflectance distribution function (BRDF). It relates pretty simply to our terms: BRDF = A * s(direction) / cos(theta) So for a Lambertian surface for example, BRDF = A / Pi. Translation between our terms and BRDF is easy.

For participation media (volumes), our albedo is usually called scattering albedo, and our scattering pdf is usually called phase function.

Chapter 4: Importance Sampling Materials Our goal over the next two chapters is to instrument our program to send a bunch of extra rays toward light sources so that our picture is less noisy. Let’s assume we can send a bunch of rays toward the light source using a pdf pLight(direction). Let’s also assume we have a pdf related to s, and let’s call that pSurface(direction). A great thing about pdfs is that you can just use linear mixtures of them to form mixture densities that are also pdfs. For example, the simplest would be: p(direction) = 0.5*pLight(direction) + 0.5*pSurface(direction) As long as the weights are positive and add up to one, any such mixture of pdfs is a pdf. And remember, we can use any pdf-- it doesn’t change the answer we converge to! So, the game is to figure out how to make the pdf larger where the product s(direction)*color(direction) is large. For diffuse surfaces, this is mainly a matter of guessing where color(direction) is high. For a mirror, s() is huge only near one direction, so it matters a lot more. Most renderers in fact make mirrors a special case and just make the s/p implicit-- our code currently does that. Let’s do simple refactoring and temporarily remove all materials that aren’t Lambertian. We can use our Cornell Box scene again, and let’s generate the camera in the function that generates the model:

At 500x500 my code produces this image in 10min on 1 core of my Macbook:

Reducing that noise is our goal. We’ll do that by constructing a pdf that sends more rays to the light. First, let’s instrument the code so that it explicitly samples some pdf and then normalizes for that. Remember MC basics: integral f(x) is approximately f(r)/p(r). For the Lambertian material, let’s sample like we do now: p(direction) = cos(theta) / Pi. We modify the base-class material to enable this importance sampling:

And Lambertian material becomes:

And the color function gets a minor modification.

You should get exactly the same picture. Now, just for the experience, try a different sampling strategy. Let’s choose randomly from the hemisphere that is above the surface. This would be p(direction) = 1/(2*Pi).

And again I should get the same picture except with different variance, but I don’t!

It’s pretty close to our old picture, but there are differences that are not noise. The front of the tall box is much more uniform in color. So I have the most difficult kind of bug to find in a Monte Carlo program-- a bug that produces a reasonable looking image. And I don’t know if the bug is the first version of the program or the second, or even in both! Let’s build some infrastructure to address this.

Chapter 5: Generating Random Directions In this and the next two chapters let’s harden our understanding and tools and figure out which Cornell Box is right. Let’s first figure out how to generate random directions, but to simplify

things let’s assume the z-axis is the surface normal and theta is the angle from the normal. We’ll get them oriented to the surface normal vector in the next chapter. We will only deal with distributions that are rotationally symmetric about z. So p(direction) = f(theta). If you have had advanced calculus, you may recall that on the sphere in spherical coordinates dA = sin(theta)*dtheta*dphi. If you haven’t, you’ll have to take my word for the next step, but you’ll get it when you take advanced calc. Given a directional pdf, p(direction) = f(theta)on the sphere, the 1D pdfs on theta and phi are: a(phi) = 1/(2Pi) (uniform) b(theta) = 2*Pi*f(theta)sin(theta) For uniform random numbers r1 and r2, the material presented in Chapter 2 leads to: r1 = INTEGRAL_0^phi (1/(2*Pi)) = phi/(2*Pi) Solving for phi we get: phi = 2*Pi*r1 For theta we have: r2 = INTEGRAL_0^theta 2*Pi*f(t)sin(t) Here, t is a dummy variable. Let’s try some different functions for f(). Let’s first try a uniform density on the sphere. The area of the unit sphere is 4Pi, so a uniform p(direction) = 1/(4Pi) on the unit sphere. r2 = -cos(theta)/2 - (-cos(0)/2)= (1 - cos(theta))/2 Solving for cos(theta) gives: cos(theta) = 1 - 2*r2 We don’t solve for theta because we probably only need to know cos(theta) anyway and don’t want needless acos() calls running around.

To generate a unit vector direction toward (theta,phi) we convert to Cartesian coordinates: x = cos(phi)*sin(theta) y = sin(phi)*sin(theta) z = cos(theta) And using the identity that cos^2 + sin^2 = 1, we get the following in terms of random (r1,r2): x = cos(2*Pi*r1)*sqrt(1 - (1-2*r2)^2) y = sin(2*Pi*r1)*sqrt(1 - (1-2*r2)^2) z = 1 - 2*r2 Simplifying a little, (1-2*r2)^2 = 1 - 4*r2 + 4*r2^2, so: x = cos(2*Pi*r1)*2*sqrt(r2*(1-r2)) y = sin(2*Pi*r1)*2*sqrt(r2*(1-r2)) z = 1 - 2*r2 We can output some of these:

And plot them for free on plot.ly (a great site with 3D scatterplot support):

On the plot.ly website you can rotate that around and see that it appears uniform. Now let’s derive uniform on the hemisphere. The density being uniform on the hemisphere means p(direction) = 1/(2*Pi) and just changing the constant in the theta equations yields: cos(theta) = 1 - r2 It is comforting that cos(theta) will vary from 1 to 0, and thus theta will vary from 0 to Pi/2. Rather than plot it, let’s do a 2D integral with a known solution. Let’s integrate cosine cubed over the hemisphere (just picking something arbitrary with a known solution). First let’s do it by hand: INTEGRAL cos^3 = 2*Pi*INTEGRAL_0^PI/2 cos^3 sin = PI/2. Now for integration with importance sampling. p(direction) = 1/(2*Pi), so we average f/p which is cos^3/ (1/(2*Pi)), and we can test this:

Now let’s generate directions with p(directions) = cos(theta) / Pi. r2 = INTEGRAL_0^theta 2*Pi*(cos(t)/Pi)*sin(t) = 1-cos^2(theta) So, cos(theta) = sqrt(1-r2) We can save a little algebra on specific cases by noting z = cos(theta) = sqrt(1-r2) x = cos(phi)*sin(theta) = cos(2*Pi*r1)*sqrt(1-z^2) = cos(2*Pi*r1)*sqrt(r2) y = sin(phi)*sin(theta) = sin(2*Pi*r1)*sqrt(1-z^2) = sin(2*Pi*r1)*sqrt(r2) Let’s also start generating them as random vectors:

We can generate other densities later as we need them. In the next chapter we’ll get them aligned to the surface normal vector.

Chapter 6: Ortho-normal Bases In the last chapter we developed methods to generate random directions relative to the z-axis. We’d like to do that relative to a surface normal vector . An ortho-normal basis (ONB) is a collection of three mutually orthogonal unit vectors. The Cartesian xyz axes are one such ONB and I sometimes forget that it has to sit in some real place with real orientation to have meaning in the real world, and some virtual place and orientation in the virtual world. A picture is a result of the relative positions/orientations of the camera and scene, so as long as the camera and scene are described in the same coordinate system, all is well. Suppose we have an origin o and cartesian unit vectors x/y/z. When we say a location is (3, -2, 7), we really are saying: Location is o + 3*x - 2*y + 7*z

If we want to measure coordinates in another coordinate system with origin o’ and basis vectors u/v/w, we can just find the numbers (u, v, w) such that: Location is o’ + uu + vv + w. If you take an intro graphics course, there will be a lot of time spent on coordinate systems and 4x4 coordinate transformation matrices. Pay attention, it’s important stuff in graphics! But we won’t need it. What we need to is generate random directions with a set distribution relative to n. We don’t need an origin because a direction is relative to no specified origin. We do need two cotangent vectors that are mutually perpendicular to n and each other. Some models will come with at least one cotangent vector. The hard case of making an ONB is when we just have one vector. Suppose we have any vector a that is not zero length or parallel to n. We can get to vectors s and t parallel to n by using the property of the cross product that cross(c,d) is perpendicular to both c and d: t = unit_vector( cross(a, n) ) s = cross(t,n) The catch is, we don’t have an a and if we pick a particular one at some point we will get an n parallel to a. A common method is to use an if statement to determine whether n is a particular axis, and if not, use that axis. If (fabs(n.x()) > 0.9) a = (0,1,0) else a = (1,0 ,0) Once we have an ONB and we have a random (x,y,z) relative the the z-axis we can get the vector relative to n as: Random vector = x*s + y*t + z*n You may notice we used similar math to get rays from a camera. That could be viewed as a change to the camera’s natural coordinate system. Should we make a class for ONBs or are

utility functions enough? I’m not sure, but let’s make a class because it won't really be more complicated than utility functions:

We can rewrite our Lambertian material using this to get:

Which produces:

Is that right? We still don’t know for sure. Tracking down bugs is hard in the absence of reliable reference solutions. Let’s table that for now and move on to get rid of some of that noise.

Chapter 7: Sampling Lights Directly The problem with sampling almost uniformly over directions is that lights are not sampled any more than unimportant directions. We could use shadow rays and separate out direct lighting. But instead, I’ll just send more rays to the light. We can then use that later to send more rays in whatever direction we want. It’s really easy to pick a random direction toward the light; just pick a random point on the light and send a ray in that direction. But we also need to know the pdf(direction). What is that?

For a light of area A, if we sample uniformly on that light, the pdf on the surface of the light is 1/A. But what is it on the area of the unit sphere that defines directions? Fortunately, there is a simple correspondence, as outlined in the diagram:

If we look at a small area dA on the light, the probability of sampling it is p_q(q)*dA. On the sphere, the probability of sampling the small area dw on the sphere is p(direction)*dw. There is a geometric relationship between dw and dA: dw = dA cos(alpha) / (distance(p,q)^2) Since the probability of sampling dw and dA must be the same, we have p(direction)*cos(alpha)*dA / (distance(p,q)^2 ) = p_q(q)*dA = dA / A So p(direction) = distance(p,q)^2 / ( cos(alpha) *A ) If we hack our color function to sample the light in a very hard-coded fashion just to check that math and get the concept, we can add it (see the highlighted region):

With 10 samples per pixel this yields:

This is about what we would expect from something that samples only the light sources, so this appears to work. The noisy pops around the light on the ceiling are because the light is two-sided and there is a small space between light and ceiling. We probably want to have the light just emit down. We can do that by letting the emitted member function of hitable take extra information:

We also need to flip the light so its normals point in the -y direction and we get:

Chapter 8: Mixture Densities

We have used a pdf related to cosine theta, and a pdf related to sampling the light. We would like a pdf that combines these. A common tool in probability is to mix the densities to form a mixture density. Any weighted average of pdfs is a pdf. For example, we could just average the two densities: pdf_mixture(direction) = ½ pdf_reflection(direction) + ½ pdf_light(direction)

How would we instrument our code to do that? There is a very important detail that makes this not quite as easy as one might expect. Choosing the random direction is simple: if (drand48() < 0.5) Pick direction according to pdf_reflection else Pick direction according to pdf_light But evaluating pdf_mixture is slightly more subtle. We need to evaluate both pdf_reflection and pdf_light because there are some directions where either pdf could have generated the direction. For example, we might generate a direction toward the light using pdf_reflection. If we step back a bit, we see that there are two functions a pdf needs to support: 1. What is your value at this location? 2. Return a random number that is distributed appropriately. The details of how this is done under the hood varies for the pdf_reflection and the pdf_light and the mixture density of the two of them, but that is exactly what class hierarchies were invented for! It’s never obvious what goes in an abstract class, so my approach is to be greedy and hope a minimal interface works, and for the pdf this implies:

We’ll see if that works by fleshing out the subclasses. For sampling the light, we will need hitable to answer some queries that it doesn’t have an interface for. We’ll probably need to mess with it too, but we can start by seeing if we can put something in hitable involving sampling the bounding box that works with all its subclasses. First, let’s try a cosine density:

We can try this in the color() function, with the main changes highlighted. We also need to change variable pdf to some other variable name to avoid a name conflict with the new pdf class.

This yields an apparently matching result so all we’ve done so far is refactor where pdf is computed:

Now we can try sampling directions toward a hitable like the light.

This assumes two as-yet not implemented functions in the hitable class. To avoid having to add instrumentation to all of hitable subclasses, we’ll add two dummy functions to the hitable class:

And we change xz_rect to implement those functions:

And then change color() :

At 10 samples per pixel we get:

Now we would like to do a mixture density of the cosine and light sampling. The mixture density class is straightforward:

And plugging it into color():

1000 samples per pixel yields:

We’ve basically gotten this same picture (with different levels of noise) with several different sampling patterns. It looks like the original picture was slightly wrong! Note by “wrong” here I mean not a correct Lambertian picture. But Lambertian is just an ideal approximation to matte, so our original picture was some other accidental approximation to matte. I don’t think the new one is any better, but we can at least compare it more easily with other Lambertian renderers.

Chapter 9: Some Architectural Decisions I won't write any code in this chapter. We’re at a crossroads where I need to make some architectural decisions. The mixture-density approach is to not have traditional shadow rays and is something I personally like, because in addition to lights you can sample windows or bright cracks under doors or whatever else you think might be bright. But most programs

branch, and send one or more terminal rays to lights explicitly and one according to the reflective distribution of the surface. This could be a time you want faster convergence on more restricted scenes and add shadow rays; that’s a personal design preference. There are some other issues with the code. The pdf construction is hard coded in the color() function and we should clean that up. Probably we should pass something into color about the lights. Unlike bvh construction, we should be careful about memory leaks as there are an unbounded number of samples. The specular rays (glass and metal) are no longer supported. The math would work out if we just made their scattering function a delta function. But that would be floating point disaster. We could either separate out specular reflections, or have surface roughness never be zero and have almost-mirrors that look perfectly smooth but don’t generate NaNs. I don’t have an opinion on which way to do it-- I have tried both and they both have their advantages-- but we have smooth metal and glass code anyway, so I add perfect specular surfaces that do not do explicit f()/p() calculations. We also lack a real background function infrastructure in case we want to add an environment map or more interesting functional background. Some environment maps are HDR (the R/G/B components are floats rather than 0-255 bytes usually interpreted as 0-1). Our output has been HDR all along and we’ve just been truncating it. Finally, our renderer is RGB and a more physically-based one-- like an automobile manufacturer might use-- would probably need to use spectral colors and maybe even polarization. For a movie renderer, you would probably want RGB. You can make a hybrid renderer that has both modes, but that is of course harder. I’m going to stick to RGB for now, but I will revisit this near the end of the book.

Chapter 10: Cleaning Up pdf Management. So far I have the color function create two hard-coded pdfs: 1. p0() related to the shape of the light

2. p1() related to the normal vector and type of surface We can pass information about the light (or whatever hitable we want to sample) into the color function, and we can ask the material function for a pdf (we would have to instrument it to do that). We can also either ask hit function or the material class to supply whether there is a specular vector. One thing we would like to allow for is a material like varnished wood that is partially ideal specular (the polish) and partially diffuse (the wood). Some renderers have the material generate two rays: one specular and one diffuse. I am not fond of branching, so I would rather have the material randomly decide whether it is diffuse or specular. The catch with that approach is that we need to be careful when we ask for the pdf value and be aware of whether for this evaluation of color() it is diffuse or specular. Fortunately, we know that we should only call the pdf value() if it is diffuse so we can handle that implicitly. We can redesign material and stuff all the new arguments into a struct like we did for hitable:

The Lambertian material becomes simpler (highlighted where the change is):

And color() changes are small:

We have not yet dealt with specular surfaces, nor instances that mess with the surface normal, and we have added a memory leak by calling new in Lambertian material. But this design is clean overall, and those are all fixable. For now, I will just fix specular. Metal is easy to fix.

Note that if fuzziness is high, this surface isn’t ideally specular, but the implicit sampling works just like it did before. Color just needs a new case to generate an implicitly sampled ray:

We also need to change the block to metal.

The resulting image has a noisy reflection on the ceiling because the directions toward the box are not sampled with more density.

We could make the pdf include the block. Let’s do that instead with a glass sphere because it’s easier. When we sample a sphere’s solid angle uniformly from a point outside the sphere, we are really just sampling a cone uniformly (the cone is tangent to the sphere). Let’s say the code has theta_max. Recall from Chapter 5, that to sample theta we have: r2 = INTEGRAL_0^theta 2*Pi*f(t)sin(t) Here f(t) is an as yet uncalculated constant C, so: r2 = INTEGRAL_0^theta 2*Pi*C*sin(t) Doing some algebra/calculus this yields:

r2 = 2*Pi*C*(1-cos(theta)) So cos(theta) = 1 - r2/(2*Pi*C) We know that for R2 = 1 we should get theta_max, so we can solve for C: cos(theta) = 1 + r2*(cos(theta_max)-1) Phi we sample like before, so: z = cos(theta) = 1 + r2*(cos(theta_max)-1) x = cos(phi)*sin(theta) = cos(2*Pi*r1)*sqrt(1-z^2) y = sin(phi)*sin(theta) = sin(2*Pi*r1)*sqrt(1-z^2) Now what is theta_max?

We can see from the figure that sin(theta_max) = R / length(c-p). So: cos(theta_max) = sqrt(1- R^2/lenghth_squared(c-p)) We also need to evaluate the pdf of directions. For directions toward the sphere this is 1/solid_angle. What is the solid angle of the sphere? It has something to do with the C above. It, by definition, is the area on the unit sphere, so the integral is

Solid_angle = INT_0^2Pi INT_0^theta_max sin(theta) = 2*Pi (1-cos(theta_max)) It’s good to check the math on all such calculations. I usually plug in the extreme cases (thank you for that concept, Mr. Horton-- my high school physics teacher). For a zero radius sphere cos(theta_max) = 0, and that works. For a sphere tangent at p, cos(theta_max) = 0, and 2*Pi is the area of a hemisphere, so that works too. The sphere class needs the two pdf-related functions:

With the utility function:

We can first try just sampling the sphere rather than the light:

This yields a noisy box, but the caustic under the sphere is good. It took five times as long as sampling the light did for my code. This is probably because those rays that hit the glass are expensive!

We should probably just sample both the sphere and the light. We can do that by creating a mixture density of their two densities. We could do that in the color function by passing a list of hitables in and building a mixture pdf, or we could add pdf functions to hitable_list. I think both tactics would work fine, but I will go with instrumenting hitable_list.

We assemble a list to pass in to color.

And we get a decent image with 1000 samples as before:

An astute reader pointed out there are some black specks in the image above. All Monte Carlo Ray Tracers have this as a main loop: pixel_color = average(many many samples) If you find yourself getting some form of acne in the images, and this acne is white or black, so one "bad" sample seems to kill the whole pixel, that sample is probably a huge number or a NaN. This particular acne is probably a NaN. Mine seems to come up once in every 10-100 million rays or so.

So big decision: sweep this bug under the rug and check for NaNs, or just kill NaNs and hope this doesn't come back to bite us later. I will always opt for the lazy strategy, especially when I know floating point is hard. First, how do we check for a NaN? The one thing I always remember for NaNs is that any if test with a NaN in it is false. This leads to the common trick:

And we can insert that in the main loop:

Happily, the black specks are gone:

Chapter 11: The Rest of Your Life The purpose of this book was to show the details of dotting all the i’s of the math on one way of organizing a physically-based renderer’s sampling approach. Now you can explore a lot of different potential paths. If you want to explore Monte Carlo methods, look into bidirectional and path spaced approaches such as Metropolis. Your probability space won't be over solid angle but will instead be over path space, where a path is a multidimensional point in a high-dimensional space. Don’t let that scare you-- if you can describe an object with an array of numbers, mathematicians call it a point in the space of all possible arrays of such points. That’s not just for show. Once you get a clean abstraction like that, your code can get clean too. Clean abstractions are what programming is all about!

If you want to do movie renderers, look at the papers out of studios and Solid Angle. They are surprisingly open about their craft. If you want to do high-performance ray tracing, look first at papers from Intel and NVIDIA. Again, they are surprisingly open. If you want to do hard-core physically-based renderers, convert your renderer from RGB to spectral. I am a big fan of each ray having a random wavelength and almost all of the RGBs in your program turning into floats. It sounds inefficient, but it isn’t! Regardless of what direction you take, add a glossy BRDF model. There are many to choose from and each has its advantages. Finally, please send me your images and fixes and complaints! I will maintain links to further reading etc. at in1weekend.com Have fun! Peter Shirley Salt Lake City, March, 2016